Wireless power transfer in the fresnel zone with a dynamic metasurface antenna

ABSTRACT

A metasurface antenna can be configured to focus a paraxial beam, such as a Gaussian beam, on a target within a Fresnel zone region. The focused beam can be used to wirelessly deliver power to the target.

All subject matter of the Priority Application(s) is incorporated hereinby reference to the extent such subject matter is not inconsistentherewith.

BACKGROUND AND SUMMARY

Wireless power transfer (WPT) has been an active topic of research, witha number of WPT schemes implemented in the near-field (coupling) andfar-field (radiation) regimes. Here, we describe a beamed WPT schemebased on a dynamically reconfigurable source aperture transferring powerto receiving devices within the Fresnel region. In this context, thedynamic aperture resembles a reconfigurable lens capable of focusingpower to a well-defined spot, whose dimension can be related to a pointspread function (PSF). In one approach, the amplitude and phasedistribution of the field imposed over the aperture can be determined ina holographic sense, by interfering a hypothetical point source locatedat the receiver location with a reference wave (e.g. a plane wave) atthe aperture location. While conventional technologies, such as phasedarrays, can achieve the required control over phase and amplitude, theytypically do so at a high cost; alternatively, metasurface apertures canachieve dynamic focusing with potentially lower cost. We present aninitial tradeoff analysis of the Fresnel region WPT concept assuming ametasurface aperture, relating key parameters such as spot size,aperture size, wavelength and focal distance, as well as reviewingsystem considerations such as the availability of sources and powertransfer efficiency. We find that approximate design formulas derivedfrom the Gaussian optics approximation provide useful estimates ofsystem performance, including transfer efficiency and coverage volume.The accuracy of these formulas is confirmed through numerical studies.

Despite the dramatic growth of wireless technology in the communicationsdomain, the use of wireless technology to provide power to devicesremains in its infancy, due to both technical as well as market relatedconcerns. Wireless power transfer (WPT) can allow power to be deliveredwithout requiring a wiring infrastructure—a useful feature especiallyfor remote, difficult to access devices, as well as embedded devices andsensors. The challenge for WPT, however, is in achieving ahigh-efficiency system at reasonable transfer distances. The dominantapproach to date for WPT has made use of magnetic near-fields, in whichpower is transferred between source and receiver coils coupled throughnon-radiating magnetic fields at very low frequencies of operation(kilohertz through megahertz, for example)^(1,2,3). Because near-fieldmagnetic WPT systems are safe in terms of human exposure and can behighly efficient at short distances, they have led to numerouscommercialization efforts⁴. However, because the near-field couplingfalls off rapidly with distance between source and receiver (as thesixth power of the inverse distance)⁵, near-field WPT schemes requirethe receiving device to be in close proximity (<3λ₀)⁶ to the powersource. While this proximity constraint is less problematic for someapplications, such as vehicle charging, it remains an inconvenience inother contexts, and can rule out entire application areas such aspowering remote sensors at long ranges.

At the other extreme, power transfer can be accomplished using shortwavelength radio frequency (RF) power radiated from a source aperture toa receive antenna or aperture^(7,8). The advantage of such a WPT systemis that power can be transferred over very long distances (from 3λ₀ tothe limit of the Fresnel zone)^(6,7) to targets at arbitrary locations,potentially in hard to access regions or embedded in other materialsthat are transparent at RF frequencies. The disadvantage for far-fieldsystems is that the beam width from an aperture is limited bydiffraction, so that only a minute fraction of power supplied by thesource is captured by the receive aperture. In far-field scenarios, forwhich the distance between source and receive apertures is greater thand=2D²/λ₀ (where D is the aperture dimension)⁹, the ratio of the powercaptured by the receiver to the supplied power is governed by the Friisequation. To achieve even modest efficiency levels for WPT schemes inthe far-field regime, enormous apertures would be required. In addition,if the target to be powered is in motion, such as might be the case foran unmanned aerial vehicle (UAV) or autonomous automobile, then thesource aperture would need to either be mechanically scanned orelectronically reconfigurable.

If the distance between source and receiver is within the Fresnel zone(also termed the “radiating near-field,” d<2D²/λ₀), and a line of sightis available, then a high efficiency WPT system can be realized by usinga large aperture that acts as a lens, concentrating electromagneticenergy at a focal point where the receiving aperture ispositioned^(10,11,12). In this scenario, a method of dynamicallycreating and moving a focal spot is needed. Recent intense research anddevelopment in the area of metasurface apertures—guided wave structuresthat radiate energy through an array of patterned irises—have shown apath to extremely low-cost, mass-producible reconfigurable aperturesthat could be applied for WPT applications^(13,14). The metasurfaceantenna is a passive device, in the sense that active phase shifters arenot required to achieve dynamic tuning. Thus, for many metasurfaceaperture implementations minimal power would need to be supplied toachieve beam steering, in contrast to typical phased array orelectronically scanned antenna systems. Through the control over thephase or amplitude of each radiating element, holographic patterns canbe created on the metasurface that mimic the functionality of Fresnellenses or other diffractive optical elements. The metasurface aperturethus effectively can function as a low-cost, dynamically reconfigurablelens that consumes minimal power. A Fresnel zone WPT system based on ametasurface aperture can thus potentially achieve very high efficiencyat minimal cost.

There are a variety of factors that must be considered for achieving aviable Fresnel zone WPT platform. In particular, the wavelength ofoperation represents a critical design choice. Ideally, short wavelengthradiation is desirable, since very small focal spot sizes can be formedwith moderate sized apertures. However, the cost of microwave sourcesincreases dramatically at shorter wavelengths, forming a crucialtradeoff decision for the system design. Currently, fairly large,dynamically reconfigurable metasurface apertures have been demonstratedat X (8-12 GHz)¹⁵ and K (18-26.5 GHz)¹⁶ bands, and are feasible athigher frequencies including V (40-75 GHz) and W (75-110 GHz) bands. WPTsystems operating within any of these bands are feasible depending onthe particular application. Power harvesting elements at thesewavelengths, such as rectennas, have been demonstrated, but highconversion efficiency (>50%)¹⁷ circuits may require additionaldevelopment. Assuming optimal conversion efficiency, we can obtainestimates of the useful range of a Fresnel zone WPT system as a functionof aperture size and frequency of operation, based on the ideal fieldpatterns expected to fill the aperture.

There are many potential usage opportunities for WPT schemes operatingin the near-zone. Since the location of the Fresnel zone (and realizablefocus spot size) depends only on the size of the transmit aperture andthe wavelength of operation, many different scenarios can be considered.One possible scenario is presented in FIG. 1, which shows an aperturebeing used to beam power wirelessly to electronic devices within theconfines of a room. The advantage of such a scheme is that devices—suchas cell phones, laptops, computer peripherals, gaming controllers orconsoles, watches, radios, small appliances—can be positioned anywherewithin the line of sight of the source to receive power, requiring nocables or charging stations. In this approach, each device might bediscoverable and locatable by some separate wireless system, which couldbe built into the protocols of a Fresnel zone WPT system. That is, eachdevice might signal its presence in the room, and communicate itslocation and orientation with respect to the transmit aperture. Inaddition, if the device is being powered, it might indicate to thetransmit aperture that power is being delivered, such that the systemshuts down if the direct path is blocked for any reason (e.g., if aperson passes between the aperture and device). This safety interlocksystem could be utilized to accommodate FCC mandated human exposurelimits. Since our goal here is to consider the general viability ofFresnel zone power transfer from a power transfer efficiencyperspective, we do not consider further the issues of safety and otherrelated protocols that would ultimately become a system engineeringtopic. We note that if an appropriate interlock system can be designedbased on two-way communication between the source and receiver, fieldsabove the regulatory limits would never be incident on any unintendedtarget.

The actual metasurface antenna implementation of a dynamic aperture willhave limitations that arise due to the finite size of the metamaterialelements (leading to a subwavelength sampling of the aperture), as wellas their inherent dispersive characteristics. Using a well-establishedmodel for these elements that describes both their dispersive andradiative properties, it is possible to determine the actual focal spotsize and shape, including aberrations introduced by any phase oramplitude limitations inherent to the elements. Phase or amplitudepatterns on the metasurface aperture that steer the focal spotthroughout the volume of coverage can be determined using holographictechniques, so the effective power transfer efficiency can be studied asa function of the receiver location and orientation.

For the purposes of the analysis presented here, we consider anillustrative example of powering devices within a room of dimensions of5 m×5 m×2.5 m, requiring an aperture large enough that all points withinthe room are also within the Fresnel region of the transmit aperture. Asshown in FIG. 1, a single transmit aperture (P1) focuses RF power toseveral electronic devices within the room, such as R1 and R2. Thedevices are placed at arbitrary locations in the room, at differentfocal depths with respect to the transmit aperture, as would be expectedin real use scenarios. The transmit aperture must thus be capable ofpowering targets at different depths and angles, as well as potentiallypowering multiple receivers from a single aperture. This functionalityimplies a dynamic aperture capable of creating a tight focus andadjustable focal length.

In an embodiment, a method is provided for operating a metasurfaceantenna, comprising configuring the metasurface antenna to focus aparaxial beam on a target within a Fresnel zone region of themetasurface antenna. In another embodiment, an apparatus is providedthat includes a metasurface antenna that is configured to focus aparaxial beam on a target within a Fresnel zone region of themetasurface antenna. In yet another embodiment, a system is providedthat includes a storage medium on which a set of antenna configurationsfor a metasurface antenna is written, each antenna configurationproviding a paraxial beam focused on a target within a Fresnel zoneregion of the metasurface antenna, the system further including controlcircuitry operable to read antenna configurations from the storagemedium and adjust the metasurface antenna to provide the antennaconfigurations.

In general, a “metasurface antenna” herein refers to an antenna thatincludes an aperture or surface that is populated with an array ofsubwavelength elements. In one approach, the subwavelength elements haveinter-element spacings that are less than one-half, one-third, orone-fifth of a free-space wavelength corresponding to an operatingfrequency of the antenna. In some approaches, the metasurface antenna isoperated by causing the individual subwavelength elements to radiateresponsive to a reference wave. The reference wave could be, forexample, a guided wave that is propagated along the aperture and coupledto the individual elements, or a free-space wave that is incident uponthe aperture and coupled to the individual elements. The metasurfaceantenna can be reconfigured, for example, by adjusting the responses ofthe individual elements to the reference wave, e.g. by adjusting controlinputs (such as bias voltage inputs) that address the elements of theaperture. Various examples of metasurface antennas are described in A.Bily et al, “Surface Scattering Antennas,” U.S. Patent ApplicationPublication No. 2012/0194399; A. Bily et al, “Surface Scattering AntennaImprovements,” U.S. Patent Application Publication No. 2014/0266946; P.-Y. Chen et al, “Surface Scattering Antennas with Lumped Elements,” U.S.Patent Application Publication No. 2015/0318618; E. Black et al,“Slotted Surface Scattering Antennas,” U.S. Patent ApplicationPublication No. 2015/0380828; and J. Bowers et al, “Surface ScatteringReflector Antenna,” U.S. Patent Publication No. 2015/0162658; each ofwhich is herein incorporated by reference.

In general, a “paraxial beam” herein refers to a beam of electromagneticradiation for which the beam propagates along a propagation axis, andthe spatial variation of the beam can be described, to a goodapproximation, as a principal phase advance along the propagation axismultiplied by a much slower variation describing the envelope of thebeam. A Gaussian beam is a specific example of a paraxial beam in whichthe beam has a Gaussian profile transverse to the propagation axis, asdescribed in greater detail below. Other examples of paraxial beamsinclude Hermite-Gaussian beams (wherein the paraxial beam may bedecomposed into an orthogonal set of Hermite-Gaussian modes in aCartesian coordinate system), Laguerre-Gaussian beams (wherein theparaxial beam may be decomposed into an orthogonal set ofLaguerre-Gaussian modes in a cylindrical coordinate system), andInce-Gaussian beams (wherein the paraxial beam may be decomposed into anorthogonal set of Ince-Gaussian modes in an elliptic coordinate system).Throughout this disclosure, when reference is made to a Gaussian beam ora Gaussian mode, embodiments are contemplated that substitute a moregeneral paraxial beam or paraxial mode for the specific Gaussian modethat is described in any particular context of the disclosure.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1. Potential usage scenario for a Fresnel zone wireless powertransfer system.

FIG. 2 Intensity plots of a focused beam from an aperture of dimensionD=1 m, at a frequency of 10 GHz, plotted for several values of focallength (z₀). Solid white curves are plots of the beam waist w(z).

FIG. 3(a) Transfer efficiency assuming the receive aperture is placed atthe focus, as a function of transmit aperture size D (scaled to the beamwaist w₀). (b) Transfer efficiency assuming a receive aperture of sizew₀, where (D/w₀=1), as a function of scaled distance away from thefocus. (1/e² points in (b) correspond to ˜−4z/z_(R) and +4 z/z_(R))

FIG. 4(a) Maximum transfer efficiency (η_(upper lim)) as a function ofminimum beam waist for several different transfer distances (D, inmeters). (b) Optimized minimum beam waist vs transfer distance, forseveral different Rx radii (in meters). Tx radius 1 m and frequency 5.8GHz are assumed. The minimum beam waist is normalized to Rx radius.

FIG. 5 1-D cross-section plot (zoomed version) of the normalizedelectric field intensity at the focal plane for on-axis focusing fordifferent aperture dimensions.

FIG. 6 Cross range 2D plots (a) Phase distribution for both on- andoff-axis case at the aperture plane (b) Focused spots (zoomed in) atfocal plane for both on- and off-axis focusing (Normalized).

FIG. 7(a) Analytical (Eq. 5) and numerical beam waist as a function ofoffset angle θ for a focal length z₀=5 m. (b) Power transfer efficiencyas a function of offset angle (D_(RX)=3 cm).

FIG. 8 Illustrative coverage plot of beam waist in cm as a function ofoff axis angle and focal distance in a room (5 m×5 m, top view).Contours of constant beam waist are computed from (a) the analyticalequation and (b) numerical simulations.

FIG. 9 Field distributions for an amplitude holographic lens designed toproduce a focus at 15° away from the normal. (a) Intensity distributionin dB of the y-z plane showing higher diffraction orders (b) Intensitydistribution in dB of the x-y plane (cross range) plot of off-axisfocusing with the zeroth order mode at the center.

FIG. 10 Intensity distributions (dB) in the y-z plane (range) and x-yplane (cross-range) for a phase holographic lens designed to produce afocus at 15° away from the normal. (a) Aperture D (b) Aperture E (c)Aperture F (d) Aperture G.

FIG. 11 Field distributions of a metasurface aperture designed toproduce a focus at 15° away from the normal. The phase variation here isrestricted to lie between −90° and +90° (aperture H in Table IV). (a)Intensity distribution in dB of the y-z plane showing higher diffractionorders (b) Intensity distribution in dB of the x-y plane (cross range)plot of off axis focusing with the zeroth order mode at the center.

FIG. 12 Illustration of the effective aperture considered for theanalytical calculation of the beam waist for off-axis focusing.

DETAILED DESCRIPTION

FIG. 13 illustrates an example table of beam waist as a function offrequency and aperture size.

In the following detailed description, reference is made to theaccompanying drawings, which form a part hereof. In the drawings,similar symbols typically identify similar components, unless contextdictates otherwise. The illustrative embodiments described in thedetailed description, drawings, and claims are not meant to be limiting.Other embodiments may be utilized, and other changes may be made,without departing from the spirit or scope of the subject matterpresented here.

Power Transfer in the Fresnel Zone

The efficiency of a WPT system in either the Fresnel (<2D²/λ₀) or theFraunhofer (>2D²/λ₀) regime depends predominantly on the effectiveaperture sizes of the source and receiver, as well as the free spacewavelength. Given our choice of Fresnel region operation, the far-fieldpropagation model is not valid. Instead, in the Fresnel region, theaperture behaves like a lens, able to concentrate the transmitted energyto a volume defined by its point spread function (PSF). As anillustrative example, we assume the aperture behaves as a Fresnel lens,and apply Gaussian optics to characterize the expected field patterns.We consider the line of sight case where a source beam will travelwithout encountering obstructions. The spatial electric fielddistribution corresponding to a Gaussian beam can be describedanalytically using the expression^(18,19)

$\begin{matrix}{{\frac{E\left( {x,y,z} \right)}{E_{0}} = {\frac{w_{0}}{w}e^{- \frac{r^{2}}{w^{2}}}e^{{- i}\frac{k_{0}r^{2}}{2R}}e^{- {i{({{k_{0}z} - \phi})}}}}},} & (1)\end{matrix}$where r=√{square root over (x²+y²)} is the distance from the center ofthe focus. E₀ is the amplitude of the electric field at the focus, k₀ isthe wavenumber for the free space wavelength λ₀ and z is the axialdistance from the position of the focus. In this expression w is thebeam waist, given by

$\begin{matrix}{{w(z)} = {w_{0}{\sqrt{1 + \left( \frac{z}{z_{r}} \right)^{2}}.}}} & (2)\end{matrix}$In Eq. 2, w₀=w (0) is the beam waist at the focus, and z_(g) is theRayleigh length, defined as

$\begin{matrix}{{z_{R} = \frac{\pi\; w_{0}^{2}}{\lambda_{0}}},} & (3)\end{matrix}$Finally, ϕ is the Gouy phase shift and R is the radius of curvature,which have expressions

$\begin{matrix}{{{{\phi(z)} = {\tan^{- 1}\left( \frac{z}{z_{R}} \right)}}{R(z)} = {z\left( {1 + \frac{z_{R}^{2}}{z^{2}}} \right)}},} & (4)\end{matrix}$Assuming that the fields are focused from a lens of diameter D and focallength z₀, the minimum beam waist can be calculated as

$\begin{matrix}{w_{0} = {\frac{4}{\pi}{\frac{z_{0}\lambda_{0}}{D\;{\cos^{2}(\theta)}}.}}} & (5)\end{matrix}$where θ is the angle between the optical axis of the aperture and thevector defined from the aperture center to the position of the focus.The cosine factor is introduced to approximately account for thedecreased effective aperture for off-axis focal positions (on-axiscorresponds to θ=0°). Eq. 5 is derived in the appendix.

Eq. 1 shows that the fields of a focused beam tend to be tightlyconfined laterally around the focus, but extend along the propagationdirection by a distance corresponding to the Rayleigh length. Thus, forshort focal lengths (z₀) relative to the aperture dimension (D), thefields tend to be confined in all dimensions; however, for larger focallengths, the Rayleigh length tends to be larger and the fields spreadout along the propagation direction. Plots of the intensity of a focusedbeam for several values of z₀ are shown in FIG. 2. For theseillustrative plots, the aperture size is chosen as D=1 m, and thefrequency as f=10 GHz (λ₀=3 cm).

The power transferred to the receiver will depend on the ratioη=D_(Rx)/w₀, which assumes all power within the overlap region betweenthe receive aperture and the beam waist is transferred. As the minimumbeam waist increases with focal length, a straightforward designconsideration for the system is that the transfer efficiency must beoptimized at the farthest desired point from the aperture. For thisstudy, we assume without limitation the smallest receive aperture tohave a dimension of D_(Rx)=3.0 cm.

Using Eq. 5, we can perform an initial study of the beam waist versusaperture dimension and frequency as a means of assessing the initialconstraints of a WPT system. Such a study is presented in Table Iillustrated in FIG. 13, which provides the minimum beam waist at a focallength of 5 m for several values of aperture size and operatingfrequency. Table I in FIG. 13 illustrates beam waist (in cm) as afunction of frequency and aperture size, using Eq. 1. Thethicker-bordered cells represent preferred situations for thisnon-limiting example. (Considering beam waists ≤3.0 cm for practicalreceivers).

Inspection of Table I shows that, as expected, larger aperture andhigher frequency provide the smallest beam waists. Given the receiveaperture size considered here (D_(Rx)=3.0 cm), there are a variety ofcombinations of transmit aperture size and frequency that shouldoptimize efficiency. Assuming a transmit aperture footprint of no largerthan 1 m², Table I shows that frequencies of 60 GHz or higher shouldprovide reasonable transfer efficiency over the distance considered. Theeventual choice of frequency will likely be determined by theavailability, cost and power conversion efficiency of the RF powersource and the rectifier or energy harvester at the receiver. Atpresent, for example, low cost solid-state sources are emerging into themarket for bands at 60 GHz and 77 GHz, due to the demand in automotiveradar and other large market applications.

The possibility of achieving even smaller beam waists at much higherfrequencies, such as THz, infrared or even visible, can also beconsidered. However, highly efficient rectennas (rectifying antennas)can be challenging to design at these frequencies²⁰, sources areexpensive and not readily available; and the power density in suchhighly collimated systems can exceed human safety limits.

The beam waist and Rayleigh length are the critical parameters for thedescription of a Gaussian beam, and can be used to generate generalscaling arguments for various quantities of interest. For example,assuming all power incident on the receive aperture is recovered, therelative size of the receive aperture to the beam waist should be theonly relevant quantity in terms of describing efficiency. FIG. 3a showshow the efficiency scales with the ratio of η=D_(Rx)/w₀. As expected,the transfer efficiency reaches a relatively high value (>80%) when thisratio is unity, and approaches 100% as the aperture area increasesbeyond unity. This curve is invariant with respect to focal length,frequency or other parameters. Likewise, the transfer efficiency dropsoff away from the focus as a function of distance along the propagationaxis, as shown in FIG. 3b . If the efficiency is plotted against theunitless parameter z/z_(R), then a universal curve results.

A couple of items should be noted here. We have adopted a fairly simpledefinition of transfer efficiency that will be applied throughout thisanalysis. It is relevant, however, to consider in a little more detailwhat might be the upper limit on possible free-space power transferefficiency within the Gaussian beam approximation, which should beagnostic to the manner in which the beam is created and absorbed.

While we do not consider in detail the properties of the receiveaperture, if the receive aperture is also electrically large (manywavelengths), then the WPT efficiency will depend also on the receiveaperture design, which may then also be approximated as being lens-likeand described by Eqs. 1-5. The upper bound estimate on transferefficiency is based on the notion that a Gaussian beam is a free-spacemode with a known power flux distribution, described entirely by itsminimum waist size w₀ and the focal point. The transmitter and receiverapertures couple energy in and out of this mode. For simplicity,consider circular apertures of radii R_(Tx,Rx), both concentric with thebeam axis and perpendicular to it. In some approaches, the receiveaperture radius or receive aperture area is a radius or area of afocusing structure (such as a lens) that received the incident power andfocuses it onto rectifying circuitry. In other approaches, the receiveraperture radius or receiver aperture area is a radius or area of a arrayof rectifying elements. Based on the energy density distribution of thebeam (found from Eq. 1), and the notion that the energy is localizedwith spatial accuracy of ˜1λ₀ (which becomes a negligibly small scale inthe large aperture limit), the coupling efficiency between an apertureand the beam cannot exceed

$\begin{matrix}{{\eta_{{Tx},{Rx}} = {1 - e^{\frac{{- 2}R_{{Tx},{Rx}}^{2}}{w^{2}{(z_{{Tx},{Rx}})}}}}},} & (6)\end{matrix}$where w(z_(Tx,Rx)) is the beam waist radius in the plane of the Tx or Rxapertures, respectively, as given by the beam waist equation (Eq. 2).Then, the upper limit on energy transfer efficiency between theseapertures is given byη_(upper lim)=η_(Tx)η_(Rx).  (7)From a practical perspective, dimensions such as the aperture diametersand the distances between them are given, and beam parameters such as w₀and center coordinate (relative to either aperture) can be optimized tomaximize the efficiency (Eq. 7). It can be easily shown that, as afunction of the focal point position, the optimum of (Eq. 7) is alwaysachieved when the focal point is co-located with the smaller of the twoapertures, typically the receiver. However, the relationship between theoptimum waist and the receiver size is less trivial. FIG. 4a shows thatthe efficiency limit (Eq. 7) usually has a maximum as a function of w₀(for a fixed R_(Rx)); this is a consequence of a trade-off between thedecreasing efficiency of beam absorption on the receiver and increasingefficiency of its creation as the minimum waist size goes up(wider-waist beams are easier to create). The optimum value of w₀ isgiven by an algebraic equation, whose numerical solution is plotted inFIG. 4b . One can see that, although the optimal beam waist radius isalways of the order of the receiver radius, the exact value of the bestw₀/R_(Rx) is not unity and depends on other geometric parameters, suchas the transfer distance and the receiver radius itself.

In some approaches, the transmitter is configured to deliver a Gaussianbeam with an optimum waist area based on the size and location of thereceiver, for example by selecting an optimum value analytically, e.g.using equation (6) above, or by selecting an optimum value empirically,by receiving feedback from the receiver about the transfer efficiency asa function of varying waist size. In other approaches, e.g. where thereceive aperture is very small, the optimum waist area is simply thediffraction-limited waist area.

In what follows, we find excellent qualitative and quantitativeagreement with the Gaussian optical approximation, which supposes afocal spot with fields that decay exponentially away from the spotcenter. In reality, the effects of diffraction and finite sampling willlead to the appearance of side lobes that surround the main beam. Across-range plot over the focal spot corresponding to an actual lenswill show additional field oscillations related to diffraction that areoutside of the Gaussian model. These additional lobes will contain asmall amount of power, and could slightly modify realistic efficiencycalculations, though not significantly. The side lobes are a simpleconsequence of filling the aperture more-or-less uniformly with aconstant field (modulated by the holographic patterning). We consider anillustrative case of on-axis focusing and present 1-D cross-range plotsof the normalized electric field intensity at the focal plane. For thisstudy, aperture sizes of 0.5 m, 1 m and 2 m are considered. As shown inFIG. 5, the width of the main lobe and the side lobe both change as afunction of the aperture size (the plot shown in FIG. 5 is zoomed intowards the center to focus more on the behavior of the main lobe andthe side lobes).

The ratio between the first side lobe level to the main lobe peak iscalculated to be around −14 dB shown by the black dotted line in FIG. 5,roughly in accordance with simple diffraction theory for a uniformlyilluminated aperture. The sampling on the focal plane and the apertureplane considered in this analysis is λ₀/2. The trade-off between sidelobe levels and main peak width are well-known⁸. The use of tapers andother methods of optimization could be applied to lower the side lobes,but already the side lobes can be expected to have minimal impact on thepower transfer. In the numerical simulations presented below, the sidelobes are present, and their effects are included in the efficiencycalculations.

Aperture Architecture

In some implementations of a near-zone WPT scheme, the source apertureis likely to take the form of a flat panel that can be wall- orceiling-mounted and fairly unobtrusive. Wall or ceiling mounting offersthe important advantage of line-of-sight propagation to most points in aroom. For conventional lenses, even in the most favorable ofcircumstances, some level of aberrations would be introduced into thebeam due to the inherent limitations of planar optics. While thecharacteristics of the dynamically reconfigurable sources consideredhere are distinct from static lenses and do not have the associatedgeometrical aberrations (as will be discussed), additional imperfectionsin spot size and other metrics can be expected, since the aperture willnecessarily be sampled discretely with components that may havelimitations in their phase or amplitude control range.

One means of forming a dynamic aperture is a phased array orelectronically scanned antenna^(21,22). A phased array consists of anarray of radiating elements, each element containing a phase shifter andpossibly an amplifier. The radiating elements are positioned atdistances of roughly half the free space wavelength apart. If fullcontrol over phase and amplitude is available, then it follows fromFourier optics that an electronically scanned antenna has the capabilityto produce any far-field pattern. However, from the standpoint of a highefficiency WPT system, phased arrays or electronically scanned antennasare not an optimal solution since each of the radiating modules requiresexternal bias power (beyond the wireless power to be transmitted). Powerconsumption in array control systems can be substantial, easilyexceeding the power being transferred to small devices.

An alternative architecture for dynamic focusing is that of themetasurface aperture^(23,24,25), which is—in contrast to electronicallyscanned antennas and phased arrays—a largely passive device thatachieves reconfigurability via dynamic tuning of metamaterialresonators.

Holographic Aperture Design

As a next level of approximation, we consider the formation of focalspots using an aperture over which any field distribution can beobtained. In this section, we assume that the aperture can be sampled asfinely as desired, so that the limitations associated with a flat andfinite aperture (and not sampling error) are explored. To determine thefields everywhere in the region of interest, the fields at the aperturecan be propagated using the angular spectrum method (ASM)¹⁸. In thismethod, a Fourier transform is taken of the fields on the aperture,resulting in a set of coefficients corresponding to an expansion inplane waves. Each of these plane wave components is then propagated agiven distance along the propagation direction, where an inverse Fouriertransform can be taken to find the field distribution over the plane.

We assume that an arbitrary field distribution in both amplitude andphase can be created over the aperture plane, to varying approximations,that will produce a focused spot. We determine the required fielddistributions by designing a hologram^(26,27)—the recorded interferencepattern between a reference beam and the scattered complex fields froman object located within the Fresnel region. To arrive at the requiredamplitude and phase distribution of the aperture field, we construct theaperture field by interfering a point source with a uniform plane wave.Taking the center of the aperture as the origin of the coordinatesystem, a point source located at the position (x₀,y₀,z₀) will producethe aperture field

$\begin{matrix}{{E\left( {x,y,0} \right)} \propto {\frac{e^{{ik}\sqrt{{({x - x_{0}})}^{2} + {({y - y_{0}})}^{2} + z_{0}^{2}}}}{\sqrt{\left( {x - x_{0}} \right)^{2} + \left( {y - y_{0}} \right)^{2} + z_{0}^{2}}}.}} & (8)\end{matrix}$Eq. 8 provides the amplitude and phase distribution needed to design aholographic pattern that will produce a point source ofdiffraction-limited extent. This initial field distribution can then beback-propagated to determine the fields everywhere in the region ofinterest. As a practical matter, we find that the amplitude variation isnot of great importance in reproducing the point source, so we use onlythe phase distribution in the following analysis. Examples of the phasehologram are shown in FIG. 6, as well as the focused spots produced bythese apertures. The hologram fringes must be faithfully reproduced tominimize aberrations, which means we must sample the aperture so as tocapture the spatial variation. For the simulations in this section, asampling of λ₀/8 was used.

Initially, we investigate focusing with an ideal holographic transmitaperture and a receive aperture placed some distance away. FIG. 6(a)shows the phase distribution required on the aperture plane (on-axis andoff-axis), with a 2D intensity plot of the focused spots produced bythese apertures. For both simulations, z₀=5 m. The offset value chosenfor off-axis focusing is for illustrative purposes only. Comparing theon-axis and off-axis focused intensity patterns, it can be seen that thespot for the off-axis elongates in the cross range direction; that is,the beam waist for the off-axis case slightly increases when compared tothe on-axis focusing. FIG. 6(b) includes focused spots (zoomed in) atthe focal plane for both on- and off-axis focusing (normalized).

The increase in beam waist for off-axis beams is expected due toaperture loss. To confirm the behavior, we compute the beam waist at thefocus for an ideal holographic aperture of dimension D_(Tx)=1 m. Thetransmit aperture is designed to produce a focus at a distance of z₀=5m. Since the selection of the frequency band mainly depends oncommercially-available sources, an operating frequency of 77 GHz(corresponding to the automotive radar band) is considered for theanalysis presented. The beam waist at the focal plane is taken as thediameter at which the intensity has decreased to 1/e² or 13.5% of itspeak value.

The beam waist as a function of off-axis angle (offset along y-axis) isshown in FIG. 7a for both the simulations and the approximate analyticalGaussian formula of Eq. 5. The coordinates of the focal points areselected such that the total distance of the focal point to the centerof the aperture is constant (r=√{square root over (x₀ ²+y₀ ²+z₀ ²)}).

Using, as a non-limiting example, a receive aperture of dimensionD_(Rx)=3 cm, we can assess power transfer efficiency for off-axis beams.As can be seen in FIG. 7b , the transfer efficiency decreases as thebeam offset angle θ increases, as expected due to spillover losses.

If one assumes a fixed receive aperture of minimum dimension (assumed tobe D_(Rx)=3 cm), then Eq. 5, as well as the numerical results, suggest acoverage area map can be formed that includes all regions of interestfor which the beam waist will be small enough for a desired level ofpower transfer efficiency. An analytical estimate of this coverage zonecan be extracted from Eq. 5 by examining contours in the y-z plane ofconstant beam waist. For a constant beam waist, Eq. 5 leads to thefollowing:

$\begin{matrix}{{y^{2} + \left( {z - \frac{d_{c}}{2}} \right)^{2}} = {\left( \frac{d_{c}}{2} \right)^{2}.}} & (9)\end{matrix}$where the constant d_(c) is

$\begin{matrix}{d_{c} = {D\frac{\pi}{4}{\frac{w_{0}}{\lambda_{0}}.}}} & (10)\end{matrix}$Eq. 9 shows that contour of constant beam waist is a circle in the y-zplane centered at z=d_(c)/2 and with diameter d_(c). Thus, for a givenaperture size and operating wavelength, a desired coverage range (d_(c))can be selected and Eq. 8 used to determine the beam waist needed. Inthe example we study here, the relevant parameters D=1 m, λ₀=4 mm, andw₀=3 cm suggest a coverage diameter of 6 m. All beam waists within agiven contour are smaller than this value, so that the least powertransfer efficiency occurs at the periphery of the coverage region.Plotting Eq. 5 results in a coverage map such as that shown in FIG. 8 a.

Since we expect the angular behavior predicted by Eq. 5 to be veryapproximate, we also directly compute the coverage map using thenumerical method described above. From this map (FIG. 8b ) we see thatthe contours of constant beam waist are somewhat more elliptical, yetnevertheless fairly close to the simple coverage map predicted by Eq. 9.

Phase and Amplitude Constrained Holograms

For the studies conducted above, we assumed that an arbitrary fielddistribution (in both amplitude and phase) could be imposed across theaperture, resulting in diffraction-limited focal spots being generated.Although point-by-point simultaneous control of the phase and amplitudeover an aperture can be achieved to a considerable extent in activeelectronically scanned antennas, such systems are not yet economicallyviable for larger-volume, low-cost applications such as the WPT scenarioenvisaged here. The dynamic metamaterial aperture provides a low-cost,manufacturable alternative platform, but comes with certain limitations.In particular, in some implementations it is not possible toindependently control the phase and amplitude of a resonator-basedmetasurface aperture. The resonance of the metamaterial resonatorelement possesses a Lorentzian relationship between the phase andamplitude, offering a constrained control space. Moreover, for a singleresonator, the maximum range of the phase is between −90° and +90°,placing an immediate limitation on the field distribution over theaperture²⁸. In practice, because the amplitude of a Lorentzian falls offsubstantially away from the 0° point, the useful phase range is likelysubstantially smaller. The possibility exists of combining both anelectric and a magnetic resonator into the same radiating elements,which would allow the full 360° phase range to be accessed; however,these Huygens surfaces⁴⁰ are considerably more difficult to design andmay be more subject to resistive losses and unwanted inter-elementcoupling. We will consider Lorentzian constrained holograms later.

In this section, we first investigate the potential performance of theholographic aperture for WPT in the presence of phase limitations.Because the dependence of the image produced by a hologram is typicallyonly weakly dependent on the magnitude, we first consider eitheramplitude-only or phase-only holograms as a basis for comparison.

We form the desired hologram in the same manner as above, conceptuallyinterfering a reference wave (in this non-limiting example, assumed tobe a plane wave) with the spherical wave from a point source at z₀ overthe aperture plane, resulting in the specification of the requiredcomplex field distribution over the aperture. An amplitude hologram canbe realized by enforcing a binary mask over the aperture, achieved bytreating each sampling point as either transparent or opaque; that is,each point on the aperture controls the amplitude in a binary fashion,introducing no phase shift. The result of the binary amplitude hologramis shown in FIG. 9a , where the hologram has been designed to produce afocal spot at an angle of 15° from the normal to the aperture at adistance of z₀=5 m in range. Not surprisingly, the aperture produces azeroth-order diffracted beam and several other diffracted beams, suchthat energy is lost from the main focus and additional hotspots arecreated in the region of interest. The locations of these additionalbeams and associated focal points can be determined using simplediffraction theory, and agree with the patterns found from thesimulation. The scenario illustrated in FIG. 9 can be considered aworst-case scenario, since no attempt was made to optimize the aperturedistribution. The amplitude mask was created from the ideal hologram bysetting points within a particular phase range to transparent, andpoints outside that phase range to opaque. Two simple binary amplitudedesigns were considered here: In Aperture A, those regions with phasebetween 0° and 90° were set to transparent (all other regions opaque),while in Aperture B those regions with phase between 0° and 45° were setto transparent (all other regions opaque). FIG. 9(b) includes anintensity distribution in dB of the x-y plane (cross range) plot ofoff-axis focusing with the zeroth order mode at the center.

The simulations were conducted over a cubic volume of dimension L=6 mcontaining a transmit aperture of size D_(Tx=)1 m, as shown in FIG. 9a .The focus was chosen to be at z₀=5 m, at an angle of 15° from theaperture normal. The fields are determined over the transmit aperture inthe manner described above, and set to zero elsewhere on the apertureplane. For the two apertures considered, A and B, the overall transferefficiency was determined, as summarized in Table II. As above, in thiscase one contributor to transfer efficiency is the ratio of the intendedreceive aperture (D_(Rx)) to that of the beam waist at focus, or η. Inaddition, we define the ratio α as the total power radiated by theaperture to the total power in the first order (desired) focused beam.Thus, the overall efficiency of power transfer to the device is η/α. Thetotal power is calculated by integrating the Poynting vector over theentire surface of the domain (at z₀=5 m). A value of α=1 indicates nopresence of higher order modes, while the presence of higher ordersresults in α>1.

TABLE II Overall efficiency of amplitude-only holograms for off- axisfocusing and ratio α for different binary holograms. Aperture Overallefficiency (%) α A: 0° to 90° 14.4 2.4 B: 0° to 45° 8.9 2.8

As shown in Table II, α is above 1 for both cases considered, and theefficiency is generally low, indicating that significant power is lostto higher order diffracted modes. With additional optimization, it ispossible to suppress some or all of the diffractive orders within anamplitude-only hologram, especially if the amplitude is allowed to takea range of values (grayscale) rather than just binary^(29,30). See, forexample, P. -Y. Chen et al, “Modulation Patterns for Surface ScatteringAntennas,” U.S. Patent Application Publication No. 2016/0149309, hereinincorporated by reference.

We next consider the formation of phase holograms, starting from theideal hologram specification and assuming the phase at each sampledpoint on the aperture can be controlled to some extent. An ideal phasehologram would allow the phase to vary from −180° to +180° (aperture C),leading to idea transfer efficiency, as shown in Table III.

For a phase only hologram, any limitation of the phase range to lessthan 360° will result in an imperfect hologram and degraded focusingperformance; moreover, the inevitable phase discontinuities that resultcan produce scattering into the higher order diffraction modes. Toanalyze the effect of limiting the phase, we consider again the case ofoff-axis focusing (an angle of 15° from the aperture normal), limitingthe phase values across the aperture to lie within a restricted range,as summarized in Table III. The simulation domain for these examples isidentical to that used for the amplitude-only hologram. In thesesimulations, where the phase of the ideal aperture is required to besmaller than the lower limit of the available phase range of theimplementation, the phase was set equal to the lower limit. Where thephase of the ideal aperture is required to be larger than the upperlimit of the available phase range, the phase was instead set equal tothe upper limit. As Table III shows, constraining the phase rangereduces the overall efficiency.

The field patterns for phase-only holograms with various phaseconstraints are shown in FIG. 10. Again, the aperture is designed tofocus at z₀=5 m at an off-axis angle of 15° from the normal to theaperture. Due to the phase discontinuity introduced on to the aperture,other diffraction orders occur, resulting in loss of power from the mainorder or focus. The scenarios considered in FIG. 10 correspond to theselected apertures summarized in Table III (except for Aperture C). Withthe full 360° of phase values, no higher diffraction orders wereobserved.

TABLE III Overall efficiency of phase-only holograms for off- axisfocusing and ratio α for different phase limited apertures. ApertureOverall efficiency (%) α C: −180° to +180° 78.1 1.00 D: −135° to +135°74.3 1.02 E: −90° to +90° 52.4 1.38 F: −60° to +60° 28.9 2.40 G: −30° to+30° 8.30 7.60

The beam waist of the main beam remains relatively constant for thevarious phase holograms, so that the drop in efficiency can beassociated with the power loss in the other diffracted orders. Theincreasing value of α in Table 3 indicates this loss. The idealphase-only hologram produces no higher diffraction orders, and thus hasa value of α=1. In aperture D, phase values between −135° to +135° areavailable, with the remaining sampling points, or pixels, set to eitherthe upper or lower bounds of the phase limits considered. Despiteaperture D having a significantly restricted phase range, the overallefficiency remains high and higher order modes are not important.Similarly, in aperture E, phase values between −90° to +90° areavailable. The value of α=1.38 corresponds to the increased presence ofunwanted diffracted beams as expected. Apertures F and G show asignificant increase in α because of the increase in the power lost tothe other orders, mainly the zeroth order (DC component), as shown inFIG. 10c and FIG. 10d respectively. The zeroth order becomes stronger asthe phase range on the aperture is reduced. Aperture G corresponds tothe worst case scenario of the phase limitations considered, resultingin the largest value of α. The holograms shown here are meant to beillustrative, and have not been optimized. Diffraction into unwantedorders can potentially be removed or at least suppressed through variousoptimization techniques³¹ such as iterative algorithms likeGerchberg-Saxton or Hybrid Input-Output³².

Lorentzian Constrained Holograms

We have so far considered the analysis of amplitude-only and phase-onlyholograms, but if we consider a practical metasurface antenna, theamplitude will be linked to the phase through the dispersion relation ofthe Lorentzian resonance. The metasurface antenna can be modeled as acollection of polarizable magnetic dipoles³³, with each dipolepossessing a polarizability of the form

$\begin{matrix}{\alpha_{m} = {\frac{i\; F\;\omega^{2}}{\omega^{2} - \omega_{0}^{2} + {i\;\gamma\;\omega}}.}} & (11)\end{matrix}$From Eq. 11, the phase of a metasurface element is related to itsresonance frequency according to

$\begin{matrix}{{{\tan(\theta)} = \frac{\omega^{2} - \omega_{0}^{2}}{\omega\;\gamma}},} & (12)\end{matrix}$where the γ is a loss term, ω₀ is the angular resonant frequency, and Fis a coupling factor, which we can set equal to unity for the discussionpresented here.

To form a desired phase distribution using metasurface elements, theresonance frequency of the element can be controlled by a variety ofmethods. However, by tuning the resonance frequency, the amplitude ofthe polarizability is also determined having the form

$\begin{matrix}{{\alpha_{m}} = {\frac{F\;\omega^{2}}{\sqrt{\left( {\omega^{2} - \omega_{0}^{2}} \right)^{2} + {\gamma^{2}\omega^{2}}}}.}} & (13)\end{matrix}$For a desired phase pattern θ over the aperture required to create afocus, determined by modifying the resonance frequency at each point, anamplitude pattern will necessarily be imposed defined by Eq. 13.Inserting Eq. 12 into Eq. 13 yields a relatively simple expressionlinking the phase and amplitude at each point:

$\begin{matrix}{{\alpha_{m}} = {\frac{F\;\omega}{\gamma}{{{\cos\;\theta}}.}}} & (14)\end{matrix}$Eq. 14 shows that the amplitude is proportional to the absolute value ofthe cosine of the phase (with 0° occurring at resonance), falling tozero at the extreme values (−90° or) +90°).

To investigate the impact of the Lorentzian constrained aperture, weagain consider the same scenario as above, with the aperture designed tofocus at z₀=5 m at an off-axis focusing (an angle of 15° from theaperture normal). Here we consider holograms formed by limiting thephase values across the aperture to lie within a restricted range, assummarized in Table IV, while the amplitude at each point is determinedfrom Eq. 14. The simulation domain for these examples is identical tothat used for the amplitude-only and phase-only holograms. As shown inTable IV, limiting the phase reduces the overall efficiency and producesother diffraction orders. The scenario considered in FIG. 11 correspondsto Aperture H. The beam waists for the apertures considered in Table IVare constant and the loss in overall efficiency is due to the power lostfrom the first order mode to the other diffracted orders. Aperture H canbe considered as the best case for a metasurface antenna (withoutfurther optimization), since it extends the full phase range of −90° to90°. The overall efficiency is around 34.2%. We then consider apertureI, in which the phase range is further restricted, resulting in furtherreduction in the overall efficiency due to the increase in the α term asshown in Table IV.

TABLE IV Overall efficiency of a metasurface for off- axis focusing andα for different aperture distributions. Aperture Overall efficiency (%)α H: −90° to +90° 34.2 1.8 I: −60° to +60° 18.5 3.2Sampling Criteria

The number of points at which the aperture plane is sampled is aparameter worthy of consideration. Each sampling point, or pixel,represents a point that requires dynamic control, necessitating aradiating element with tuning mechanism and an integrated bias/controlcircuit. If we consider λ₀/8 sampling for an area of 1×1 m², severalmillions of metasurface elements and associated circuitry would benecessary. We can arrive at an estimate of the required sampling byconsidering the analytic form of the fringes over the aperture at theirmost extreme spatial variation.

We consider the two-dimensional case (fields and sources invariant alongthe x-direction) to simplify the math. The interference, or fringepattern, on the aperture from a line source located at (y₀,z₀) has theform:

$\begin{matrix}{{\cos\left( {k\sqrt{\left( {y - y_{0}} \right)^{2} + z_{0}^{2}}} \right)}.} & (15)\end{matrix}$

We are interested in the spatial variation at the most extreme portionof the interference pattern, which is the farthest away from the centralspot. For the off-axis hologram, the most rapid variation will occur atthe edge of the aperture, or at:

$\begin{matrix}{{y = {{- \frac{D}{2}} + {\Delta\; y}}},} & (16)\end{matrix}$where Δy is the distance away from point at the edge of the aperture.Substituting Eq. 16 into Eq. 15:

$\begin{matrix}{{\cos\left( {k\sqrt{\left( {{\Delta\; y} - \frac{D}{2} - y_{0}} \right)^{2} + z_{0}^{2}}} \right)}.} & (17)\end{matrix}$Since Δy is smaller than any of the other quantities, we can take asmall argument expansion to obtain

$\begin{matrix}{{\cos\left( {{kz}_{0}\left( {1 + {\frac{\left( {\frac{D}{2} + y_{0}} \right)}{z_{0}^{2}}\Delta\; y}} \right)} \right)}.} & (18)\end{matrix}$A full cycle for one fringe occurs over a distance of

$\begin{matrix}{{\Delta\; y} = {\frac{\lambda_{0}z_{0}}{\frac{D}{2} + y_{0}}.}} & (19)\end{matrix}$Using Eq. 19 with z₀=1 m, λ₀=0.0039 m, D=1 m and y₀=1.34 m, we obtainΔy=0.0021 m, which is fairly close to λ₀/2 (0.0019 m). This analysissuggests that we can sample the aperture at a spacing of λ₀/2, reducingthe number of metasurface elements by a factor of 16. A focus near theaperture and at a fairly significant angle from the normal was chosen,since the spatial variation of the fringe pattern is most rapid for suchfocal spots.Microwave and Millimeter Wave Sources and Energy Harvesters

The development of low cost microwave and millimeter wave sources anddetectors has been driven by the growth of high volume markets inwireless communication and automotive radar. Unlike traditional defensemarkets, where performance is the primary criterion, cost sensitiveconsumer markets drive the development of standard components that maybe useful in beamed WPT applications. In particular, the cominggeneration of 5G wireless networks will depend on low cost, highlyintegrated silicon RF solutions. Silicon (CMOS) RF integrated circuits(RFICs) have been demonstrated well into the millimeter wave bands usingthe latest generation of sub-45 nm CMOS process technologies, wheretransistor FT can exceed 200 GHz^(35,36). Leveraging scaled CMOSprocesses, millimeter wave RF synthesizers and low noise amplifiers canbe integrated with complex digital control systems on the samesemiconductor die.

One limitation of CMOS millimeter wave technology is the limited powerhandling capability of silicon CMOS FETs. 45 nm CMOS transistorstypically have a drain breakdown voltage limit of approximately 1.1V anda peak drain power density of 100 mW/mm. Coupled with the relatively lowthermal conductivity of silicon (149 W/m-K), these limitations suggestthat alternative semiconductor technologies have an important role toplay in WPT sources. The current generation of gallium nitride (GaN) onsilicon carbide (SiC) high electron mobility transistors (HEMTs) excelfrom the perspective of high drain voltage and excellent power density.Current GaN HEMT technology supports a power density of up to 5 W/mm,over 50× greater than silicon CMOS devices. GaN power amplifiers havebeen demonstrated at power levels over 100 W in the K-band (17-26GHz)³⁷, albeit with power added efficiency of only 25-30%. Future GaNdevices with optimized geometry, configured for narrowband WPTapplications, will likely improve significantly on this starting point.It should be noted that vacuum electronics, such as gyrotron tubes, canbe very efficient in the high power regime (up to 100 kW or more), buttheir reliability and bulky high voltage power supply requirementslikely render vacuum electronics unsuitable for consumer applications.

Integrated antennas with RF energy harvesters (rectennas) based on lowcost Schottky diodes are a well-established area of research²⁰. Withcareful attention to packaging, Schottky diodes are useful up to 0.1 THzor beyond. Schottky diodes integrated with rectenna elements have beendemonstrated using on-chip antenna systems that enable a small size andrelatively high efficiency—for example, efficiency of 53% at 35 GHz and37% at 94 GHz have been reported using on-chip rectennas³⁸.

CONCLUSION

A beamed wireless power transfer (WPT) system is not without challenges,but presents an interesting alternative to near-field magnetic couplingschemes. In particular, the possibility of selectively beaming power tosmall devices located anywhere within a volume is a desirable advantage.Emerging beam-steering technologies, such as the metasurface apertureanalyzed here, have the potential to reach the low price points requiredfor larger-scale adoption in consumer driven markets. Two relevantexamples are the mTenna, currently manufactured by Kymeta Corporation(Redmond, Wash.), which is currently being commercialized as adynamically reconfigurable antenna for satellite communications, as wellas the MESA radar produced by Echodyne Corporation (Bellevue,Wash.)—both variants of the metasurface aperture. The mTenna makes useof liquid crystal as a means of implementing dynamic tuning, mergingdisplay technologies with the metasurface architecture to achievelargescale, reproducible manufacturing at low price points. The MESAmakes use of packaged active, semiconductor components integrated intothe metasurface structure to achieve dynamic tuning. Both systems haveshown that mature, conventional manufacturing solutions are viable toproduce the types of holographic apertures that would be needed for theFresnel zone WPT system described here.

For the studies pursued here, an ideal holographic metasurface aperturewith plane wave illumination was considered. A practical system wouldlikely make use of a guided mode rather than a free space wave to createa low-profile device as pictured in FIG. 1. For such guided waveimplementations, higher order diffracted beams can be further rejectedbecause of the natural phase shift of the reference wave over theaperture³⁹.

The end-to-end efficiency of a Fresnel-zone WPT system depends on threemajor factors: the RF source efficiency, the aperture and couplingefficiency analyzed here, and the RF-to-DC conversion efficiency of thereceiving energy harvester. At K-band, current efficiency forcommercially available solid state sources (˜30%) and energy harvesters(˜53%) limits end-to-end efficiency to around 15%. Assuming an 80%efficient aperture, end-to-end efficiency of around 10-12% seemsfeasible with current technology. While the efficiencies of theconstrained apertures considered here may be lower than would bedesirable for a commercial system, the phase-, amplitude- andLorentzian-constrained holograms provide basic trends and suggest animportant area of future research. By carefully optimizing the apertureto refine the phase/amplitude distributions, considerable improvementcan be made to the overall efficiency and reduction of unwanted focalspots and diffracted beams.

While we have presented simulations for one off-axis focusing scenario,we expect that the trends found for the efficiencies and other metricsare representative for all focal spots, because the beam waists of thefocal points over the entire simulation domain (for holograms withdifferent focal lengths) do not change with limitations to the phase oramplitude distribution, and are very close to that predicted by Gaussianoptics. The degradation in efficiency due to diffracted orders isroughly similar for focal points at any position.

Some embodiments provide a system that includes a metasurface antennacoupled to control circuitry operable to adjust the surface scatteringto any particular antenna configuration. The system optionally includesa storage medium on which is written a set of pre-calculated antennaconfigurations. For example, the storage medium may include a look-uptable of antenna configurations indexed by some relevant operationalparameters of the antenna, such as target location, target type, beamwaist area, etc., each stored antenna configuration being previouslycalculated. Then, the control circuitry would be operable to read anantenna configuration from the storage medium and adjust the antenna tothe selected, previously-calculated antenna configuration.Alternatively, the control circuitry may include circuitry operable tocalculate an antenna configuration according to one or more of theapproaches described above, and then to adjust the antenna for thepresently-calculated antenna configuration.

Appendix: Effective Aperture Size for Off Axis Focus

Since we are creating an ideal hologram over the aperture for everyfocal position in the range of coverage, there are no aberrationsintroduced to the focus. The widening of the beam waist must ariseentirely from the loss of aperture, which we can estimate from thegeometry. Consider the situation depicted in FIG. 12.

The thicker solid line represents the actual aperture. However, we canconceptually replace this aperture, which makes an angle θ with respectto the position of the focal spot, by a second aperture, represented bythe thinner line, for which the focal spot is now on-axis. Thecharacteristics of the focal spot must be the same for either aperture,since we assume the hologram is ideal for both cases. Given the positionof the focal spot, we have the following relationships:

$\begin{matrix}{{{\cos\;\theta} = \frac{z_{0}}{\sqrt{x_{0}^{2} + z_{0}^{2}}}}{{\tan\;\alpha} = \frac{z_{0}}{x_{0} + \frac{D}{2}}}{{{\tan\;\beta} = \frac{x_{0} - \frac{D}{2}}{z_{0}}},}} & \left( {A\; 1} \right)\end{matrix}$and R=√{square root over (x₀ ²+z₀ ²)}. We seek the length of theeffective aperture. Designating the lengths l₁ and l₂ for the twosections (on either side) of the effective aperture, we can apply thelaw of sines as follows:

$\begin{matrix}{{\frac{\sin\;\alpha}{l_{1}} = \frac{\sin\left( {\alpha - \theta} \right)}{D/2}}{{\frac{\sin\left( {\frac{\pi}{2} - \beta} \right)}{l_{2}} = \frac{\sin\left( {\frac{\pi}{2} + \beta - \theta} \right)}{D/2}},}} & \left( {A\; 2} \right)\end{matrix}$which yields

$\begin{matrix}{{l_{1} = {\frac{D}{2}\frac{\sin\;\alpha}{\sin\;\left( {\alpha + \theta} \right)}}}{{l_{2} = {\frac{D}{2}\frac{\cos\;\beta}{\cos\;\left( {\beta - \theta} \right)}}},}} & \left( {A\; 3} \right)\end{matrix}$Thus,

$\begin{matrix}{{D_{eff} = {{l_{1} + l_{2}} = {\frac{D}{2}\left\lbrack {\frac{1}{{\cos\;\theta} + {\sin\;\theta\;\cot\;\alpha}} + \frac{1}{{\cos\;\theta} + {\sin\;\theta\;\tan\;\beta}}} \right\rbrack}}},} & \left( {A\; 4} \right)\end{matrix}$While this formula for the effective aperture is not particularlyilluminating, we can rearrange the formula to find, after some algebra,

$\begin{matrix}{{D_{eff} = {{Dz}_{0}\left\lbrack \frac{R^{3}}{R^{4} + \frac{D^{2}x_{0}^{2}}{4}} \right\rbrack}},{or},} & ({A5}) \\{D_{eff} = {D\;\cos\;{{\theta\left\lbrack \frac{1}{1 + {\frac{1}{4}\left( \frac{D}{R} \right)^{2}\sin^{2}\theta}} \right\rbrack}.}}} & \left( {A\; 6} \right)\end{matrix}$Eq. A6 shows that away from the aperture, where R>>D, the aperturereduction goes simply as the cosine of the angle between the apertureaxis and the focal position. Closer to the aperture, however, theaperture reduction occurs more quickly. For the off-axis Gaussian beamwaist, then, we should use the effective aperture. Note that for theeffective aperture, the focal length z₀ is equal to R, so that

$\begin{matrix}{{w_{0} = {{\frac{4}{\pi}\frac{\lambda_{0}}{D}z_{0}} = {{\frac{4}{\pi}\frac{\lambda_{0}}{D_{eff}}R} = {{\frac{4}{\pi}\frac{\lambda_{0}}{D_{eff}}\frac{R}{z_{0}}z_{0}} = {\frac{4}{\pi}\frac{\lambda_{0}}{D_{eff}\cos\;\theta}z_{0}}}}}},{or},} & \left( {A\; 7} \right) \\{w_{0} = {\frac{4}{\pi}{{\frac{\lambda_{0}z_{0}}{D\;\cos^{2}\theta}\left\lbrack {1 + {\frac{1}{4}\left( \frac{D}{R} \right)^{2}\sin^{2}\theta}} \right\rbrack}.}}} & \left( {A\; 8} \right)\end{matrix}$If we can neglect the term in brackets—a good approximation for thecases under consideration—we obtain a simple formula for the beam waistvalid for off-axis focusing:

$\begin{matrix}{w_{0} = {\frac{4}{\pi}\frac{\lambda_{0}z_{0}}{D\;\cos^{2}\theta}}} & \left( {A\; 9} \right)\end{matrix}$as used in the text above.

REFERENCES

¹A. Sample, D. Meyer, and J. R. Smith, “Analysis, experimental results,and range adaptation of magnetically coupled resonators for wirelesspower transfer”, Ind. Electron, IEEE Trans. on 58, 544, (2011).

²A. Kurs et al., “Wireless power transfer via strongly coupled magneticresonances”, Science 317, 83, (2007).

³F. Jolani, Y. Yu, and Z. Chen, “A planar magnetically coupled resonantwireless power transfer system using printed spiral coils.” Antennas andWireless Propag. Lett, IEEE 13, 1648, (2014).

⁴S. Lee et al., “On-line electric vehicle using inductive power transfersystem.” IEEE Energy Conversion Congress and Exposition, IEEE, 2010, pp.1598-1601.

⁵G. Lipworth et al., “Magnetic metamaterial superlens for increasedrange wireless power transfer,” Sci. Rep., 4, 1, (2014).

⁶ D. Slater “Near-field antenna measurements”, Artech House, Mass.,1991.

⁷M. Xia, and S. Aim., “On the Efficiency of Far-Field Wireless PowerTransfer”, IEEE Tran. on Signal Process, 63, 2835, (2015).

⁸W. C. Brown, E. E Eves, “Beamed Microwave Power Transmission and itsApplication to Space,” IEEE Trans. Microwave Theory and Techniques, 40,1239, (1992).

⁹A. C. Balanis, Advanced engineering electromagnetics. John Wiley &Sons, (2012).

¹⁰V. R. Gowda et al., “Wireless Power Transfer in the RadiativeNear-Field”, IEEE Antennas and Wireless Propagation Letters (2016).

¹¹J. W. Sherman, “Properties of focused apertures in the Fresnelregion,” IEEE Trans. Antennas Propag., 10, 399, (1962).

12Buffi, A., P. Nepa, and G. Manara, “Design criteria fornear-field-focused planar arrays.” IEEE Antennas and PropagationMagazine 54.1 (2012): 40-50.

¹³M. Johnson et al., “Sidelobe canceling for reconfigurable holographicmetamaterial antenna.” Antennas and Propagation, IEEE Transactions on63, 1881, (2015).

¹⁴D. Sievenpiper et al., “Two-dimensional beam steering using anelectrically tunable impedance surface.” Antennas and Propagation, IEEETransactions on 51, 2713, (2003).

¹⁵D. Bouyge et al., “Reconfigurable 4 pole bandstop filter based onRF-MEMS-loaded split ring resonators,” in Proceedings of the IEEE MTT-SInternational Microwave Symposium (MTT '10), May 2010, pp. 588-591.

¹⁶T. Sleasman et al., “Dynamic metamaterial aperture for microwaveimaging.” Applied Physics Letters 107, 204104, (2015).

¹⁷S. Ladan, A. B. Guntupalli, and K. Wu. “A high-efficiency 24 GHzrectenna development towards millimeter-wave energy harvesting andwireless power transmission.” IEEE Transactions on Circuits and SystemsI: Regular Papers 61, 3358(2014).

¹⁸M. Born and E. Wolf, Principles of optics: electromagnetic theory ofpropagation, interference and diffraction of light. CUP Archive, 2000.

¹⁹J. W. Goodman, Introduction to Fourier optics, Roberts and CompanyPublishers, 2005.

²⁰C. R. Valenta and G. D. Durgin, “Harvesting Wireless Power: Survey ofEnergy-Harvester Conversion Efficiency in Far-Field, Wireless PowerTransfer Systems,” in IEEE Microwave Magazine, 15, 108, (2014).

²¹F. C. Williams and W. H. Kummer, “Electronically scanned antenna.”U.S. Pat. No. 4,276,551 (30 Jun. 1981).

²²R. C. Hansen, “Phased array antennas” John Wiley and Sons, 2009.

²³C. L Holloway, et al. “An overview of the theory and applications ofmetasurfaces: The two-dimensional equivalents of metamaterials.” IEEEAntennas and Propagation Magazine 54,10, (2012)

²⁴J. B. Pendry, D. Schurig, and D. R. Smith. “Controllingelectromagnetic fields.” Science 312, 1780, (2006).

²⁵H. Chen, A. J. Taylor, and N. Yu. “A review of metasurfaces: physicsand applications.” arXiv preprint arXiv:1605.07672(2016).

²⁶D. Gabor, “A new microscopic principle” Nature 161, 777, (1948).

²⁷H. Parameswaran, “Optical holography: principles, techniques andapplications” Cambridge University Press, 1996.

²⁸D. R. Smith et al., “Composite medium with simultaneously negativepermeability and permittivity.” Physical review letters 84, 4184 (2000).

²⁹A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generatedby computer.” Applied Optics 6, 1739, (1967).

³⁰E. Zhang et al., “Gradual and random binarization of gray-scaleholograms.” Applied optics 34, 5987, (1995).

³¹R. W. Gerchberg and W. O. Saxton, “A practical algorithm for thedetermination of phase from image and diffraction plane pictures,” Optik(Stuttg.) 35, 237, (1972).

³²J. R. Fineup, “Phase retrieval algorithms: a comparison.” AppliedOptics 21, 2758, (1982).

³³G. Lipworth et al., “Metamaterial apertures for coherent computationalimaging on the physical layer.” JOSA A 30, 1603, (2013).

³⁴LPKF U3 datasheet:http://www.lpkfusa.com/datasheets/prototyping/317-lpkf-protolaser-u3.pdf

³⁵U. Gogineni, J. A. del Alamo and C. Putnam, “RF power potential of 45nm CMOS technology,” 2010 Topical Meeting on Silicon MonolithicIntegrated Circuits in RF Systems (SiRF), New Orleans, La., 2010, pp.204-207.

³⁶R. S. Pengelly et al., “A Review of GaN on SiC High Electron-MobilityPower Transistors and MMICs,” in IEEE Transactions on Microwave Theoryand Techniques, 60, 1764, (2012).

³⁷Qorvo Spatium 30-31 GHz 120 W Ka Band GaN SSPA,http://www.qorvo.com/spatium

³²H. K. Chiou and I. S. Chen, “High-Efficiency Dual-Band On-ChipRectenna for 35- and 94-GHz Wireless Power Transmission in 0.13-μm CMOSTechnology,” in IEEE Transactions on Microwave Theory and Techniques,58, 3598, (2010).

³⁹G. Lipworth et al., “Phase and magnitude constrained metasurfaceholography at W-band frequencies,” Opt. Express 24, 19372 (2016).

⁴⁰C. Pfeiffer, A. Grbic, “Metamaterial Huygens’ surfaces: Tailoring wavefronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).

The foregoing detailed description has set forth various embodiments ofthe devices and/or processes via the use of block diagrams, flowcharts,and/or examples. Insofar as such block diagrams, flowcharts, and/orexamples contain one or more functions and/or operations, it will beunderstood by those within the art that each function and/or operationwithin such block diagrams, flowcharts, or examples can be implemented,individually and/or collectively, by a wide range of hardware, software,firmware, or virtually any combination thereof. In one embodiment,several portions of the subject matter described herein may beimplemented via Application Specific Integrated Circuits (ASICs), FieldProgrammable Gate Arrays (FPGAs), digital signal processors (DSPs), orother integrated formats. However, those skilled in the art willrecognize that some aspects of the embodiments disclosed herein, inwhole or in part, can be equivalently implemented in integratedcircuits, as one or more computer programs running on one or morecomputers (e.g., as one or more programs running on one or more computersystems), as one or more programs running on one or more processors(e.g., as one or more programs running on one or more microprocessors),as firmware, or as virtually any combination thereof, and that designingthe circuitry and/or writing the code for the software and or firmwarewould be well within the skill of one of skill in the art in light ofthis disclosure. In addition, those skilled in the art will appreciatethat the mechanisms of the subject matter described herein are capableof being distributed as a program product in a variety of forms, andthat an illustrative embodiment of the subject matter described hereinapplies regardless of the particular type of signal bearing medium usedto actually carry out the distribution. Examples of a signal bearingmedium include, but are not limited to, the following: a recordable typemedium such as a floppy disk, a hard disk drive, a Compact Disc (CD), aDigital Video Disk (DVD), a digital tape, a computer memory, etc.; and atransmission type medium such as a digital and/or an analogcommunication medium (e.g., a fiber optic cable, a waveguide, a wiredcommunications link, a wireless communication link, etc.).

In a general sense, those skilled in the art will recognize that thevarious aspects described herein which can be implemented, individuallyand/or collectively, by a wide range of hardware, software, firmware, orany combination thereof can be viewed as being composed of various typesof “electrical circuitry.” Consequently, as used herein “electricalcircuitry” includes, but is not limited to, electrical circuitry havingat least one discrete electrical circuit, electrical circuitry having atleast one integrated circuit, electrical circuitry having at least oneapplication specific integrated circuit, electrical circuitry forming ageneral purpose computing device configured by a computer program (e.g.,a general purpose computer configured by a computer program which atleast partially carries out processes and/or devices described herein,or a microprocessor configured by a computer program which at leastpartially carries out processes and/or devices described herein),electrical circuitry forming a memory device (e.g., forms of randomaccess memory), and/or electrical circuitry forming a communicationsdevice (e.g., a modem, communications switch, or optical-electricalequipment). Those having skill in the art will recognize that thesubject matter described herein may be implemented in an analog ordigital fashion or some combination thereof.

All of the U.S. patents, U.S. patent application publications, U.S.patent applications, foreign patents, foreign patent applications andnon-patent publications referred to in this specification and/or listedin any Application Data Sheet, are incorporated herein by reference, tothe extent not inconsistent herewith.

While various aspects and embodiments have been disclosed herein, otheraspects and embodiments will be apparent to those skilled in the art.The various aspects and embodiments disclosed herein are for purposes ofillustration and are not intended to be limiting, with the true scopeand spirit being indicated by the following claims.

The invention claimed is:
 1. A method of operating a metasurfaceantenna, comprising: configuring the metasurface antenna to focus aparaxial beam on a target within a Fresnel zone region of themetasurface antenna.
 2. The method of claim 1, further comprising:delivering electromagnetic energy to the target with the focusedparaxial beam.
 3. The method of claim 1, wherein the configuring of themetasurface antenna to focus the paraxial beam on the target within theFresnel zone region is a configuring of the metasurface antenna to focusa paraxial beam on a target within a selected coverage region.
 4. Themethod of claim 3, wherein the coverage region is a subregion of theFresnel zone region.
 5. The method of claim 3, wherein the coverageregion is a region wherein a diffraction-limited waist area for thefocused paraxial beam is less than or equal to a selected thresholdwaist area for the focused paraxial beam.
 6. The method of claim 3,further comprising: conditionally delivering or not deliveringelectromagnetic energy to the target with the focused paraxial beambased on whether the target is respectively located inside or outside ofthe coverage region.
 7. The method of claim 1, wherein: the configuringof the metasurface antenna to focus the paraxial beam on the target is aconfiguring to focus the paraxial beam with a selected waist area. 8.The method of claim 7, wherein the selected waist area is equal to areceive aperture area of the target.
 9. The method of claim 7, whereinthe selected waist area is an optimal waist area that optimizes atransfer efficiency between the metasurface antenna and the target. 10.The method of claim 1, wherein the configuring of the metasurfaceantenna to focus the paraxial beam is a configuring of an aperture ofthe metasurface antenna to define a hologram across the aperture. 11.The method of claim 10, wherein: the configuring of the aperture todefine the hologram is a configuring of a plurality of subwavelengthelements spanning the aperture with a modulation pattern thatcorresponds to the hologram.
 12. The method of claim 11, furthercomprising: determining the modulation pattern.
 13. The method of claim12, wherein the determining of the modulation pattern includes:identifying a reference field distribution over the aperture;identifying a transmit field distribution over the aperture thatcorresponds to the focused paraxial beam backwards propagated to theaperture; and determining the hologram as an interference patternbetween the reference field distribution and the transmit fielddistribution.
 14. The method of claim 12, further comprising: storingthe determined modulation pattern in a storage medium.
 15. The method ofclaim 11, further comprising: retrieving the modulation pattern from astorage medium.
 16. The method of claim 10, wherein the hologram is ahologram having a Lorentzian relationship between amplitude and phase.17. A system, comprising: a storage medium on which a set of antennaconfigurations for a metasurface antenna is written, each antennaconfiguration providing a paraxial beam focused on a target within aFresnel zone region of the metasurface antenna; and control circuitryoperable to read the antenna configurations from the storage medium andadjust the metasurface antenna to provide the antenna configurations.18. The system of claim 17, further comprising: the metasurface antenna.19. The system of claim 17, wherein: each antenna configurationproviding a paraxial beam focused on a target is a configurationproviding a paraxial beam focused on a target within a selected coverageregion.
 20. The system of claim 19, wherein the coverage region is asubregion of the Fresnel zone region.
 21. The system of claim 19,wherein the coverage region is a region wherein a diffraction-limitedwaist area for the focused paraxial beam is less than or equal to aselected threshold waist area for the focused paraxial beam.
 22. Thesystem of claim 17, wherein: the control circuitry is operable to selectan antenna configuration from the storage medium and adjust themetasurface antenna to provide the selected antenna configuration, wherethe selected antenna configuration provides a paraxial beam focused onthe target with a selected waist area.
 23. The system of claim 22,wherein the selected waist area is equal to a receive aperture area ofthe target.
 24. The system of claim 22, wherein the selected waist areais an optimal waist area that optimizes a transfer efficiency betweenthe metasurface antenna and the target.
 25. The system of claim 17,wherein: each antenna configuration providing a paraxial beam focused ona target is a configuration of an aperture of the metasurface antenna todefine a hologram across the aperture.
 26. The system of claim 25,wherein: each configuration of the aperture to define the hologram is aconfiguration of a plurality of subwavelength elements spanning theaperture with a modulation pattern that corresponds to the hologram. 27.The system of claim 26, wherein each modulation pattern is determined byan algorithm that includes: identifying a reference field distributionover the aperture; identifying a transmit field distribution over theaperture that corresponds to the focused paraxial beam backwardspropagated to the aperture; and determining the hologram as aninterference pattern between the reference field distribution and thetransmit field distribution.
 28. The system of claim 25, wherein eachconfiguration of the aperture to define a hologram across the apertureis a configuration of the aperture to define a hologram having aLorentzian relationship between amplitude and phase.
 29. An apparatus,comprising: a metasurface antenna configured to focus a paraxial beam ona target within a Fresnel zone region of the metasurface antenna. 30.The apparatus of claim 29, wherein the metasurface antenna is configuredto focus the paraxial beam on the target within a selected coverageregion.
 31. The apparatus of claim 30, wherein the coverage region is asubregion of the Fresnel zone region.
 32. The apparatus of claim 30,wherein the coverage region is a region wherein a diffraction-limitedwaist area for the focused paraxial beam is less than or equal to aselected threshold waist area for the focused paraxial beam.
 33. Theapparatus of claim 29, wherein: the metasurface antenna is configured tofocus the paraxial beam on the target with a selected waist area. 34.The apparatus of claim 33, wherein the selected waist area is equal to areceive aperture area of the target.
 35. The apparatus of claim 33,wherein the selected waist area is an optimal waist area that optimizesa transfer efficiency between the metasurface antenna and the target.36. The apparatus of claim 29, wherein the metasurface antenna includesan aperture, and the metasurface antenna is configured to define ahologram across the aperture.
 37. The apparatus of claim 36, wherein themetasurface antenna includes a plurality of subwavelength elementsspanning the aperture, and the subwavelength elements are configuredwith a modulation pattern that corresponds to the hologram.
 38. Theapparatus of claim 37, wherein the modulation pattern is determined byan algorithm that includes: identifying a reference field distributionover the aperture; identifying a transmit field distribution over theaperture that corresponds to the focused paraxial beam backwardspropagated to the aperture; and determining the hologram as aninterference pattern between the reference field distribution and thetransmit field distribution.